Geometric integration of non-autonomous Hamiltonian problems

Symplectic integration of autonomous Hamiltonian systems is a well-known field of study in geometric numerical integration, but for non-autonomous systems the situation is less clear, since symplectic structure requires an even number of dimensions. We show that one possible extension of symplectic methods in the autonomous setting to the non-autonomous setting is obtained by using canonical transformations. Many existing methods fit into this framework. We also perform experiments which indicate that for exponential integrators, the canonical and symmetric properties are important for good long time behaviour. In particular, the theoretical and numerical results support the well documented fact from the literature that exponential integrators for non-autonomous linear problems have superior accuracy compared to general ODE schemes.Comment: 20 pages, 3 figure

Orientation dependent pinning of (sub)grains by dispersoids during recovery and recrystallization in an Al-Mn alloy

The recrystallized grain size and texture in alloys can be controlled via the microchemistry state during thermomechanical processing. The influence of concurrent precipitation on recovery and recrystallization is here analyzed by directly correlating (sub)grains of P, CubeND or Cube orientation with second-phase particles in a cold-rolled and non-isothermally annealed Al-Mn alloy. The recrystallized state is dominated by coarse elongated grains with a strong P, weaker CubeND and even weaker Cube texture. The correlated data enables orientation dependent quantification of the density and size of dispersoids on sub-boundaries and subgrains in the deformation zones around large constituent particles. A new modified expression for the Smith-Zener drag from dispersoids on sub-boundaries is derived and used. The results show that the drag on (sub)grain boundaries from dispersoids is orientation dependent, with Cube subgrains experiencing the highest drag after recovery and partial recrystallization. The often observed size advantage of Cube subgrains in Al alloys is not realized due to the increased drag, thereby promoting particle-stimulated nucleation (PSN). Relatively fewer and larger dispersoids in deformation zones around large particles give a reduced Smith-Zener drag on PSN nuclei, thus further strengthening the effect of PSN. Observations substantiating the stronger P texture compared to the CubeND texture are a higher frequency of P subgrains and a faster growth of these subgrains. The applied methodology enables a better understanding of the mechanisms behind the orientation dependent nucleation and growth behavior during recovery and recrystallization with strong concurrent precipitation in Al-Mn alloys. In particular, the methodology gives new insights into the strong P and CubeND textures compared to the Cube texture

Correlated subgrain and particle analysis of a recovered Al-Mn alloy by directly combining EBSD and backscatter electron imaging

Correlated analysis of (sub)grains and particles in alloys is important to understand transformation processes and control material properties. A multimodal data fusion workflow directly combining subgrain data from electron backscatter diffraction (EBSD) and particle data from backscatter electron (BSE) images in the scanning electron microscope is presented. The BSE images provide detection of particles smaller than the applied step size of EBSD down to 0.03 $\mu$m in diameter. The workflow is demonstrated on a cold-rolled and recovered Al-Mn alloy, where constituent particles formed during casting and dispersoids formed during subsequent heating affect recovery and recrystallization upon annealing. The multimodal dataset enables statistical analysis including subgrains surrounding constituent particles and dispersoids' location with respect to subgrain boundaries. Among the subgrains of recrystallization texture, Cube{001}\left subgrains experience an increased Smith-Zener drag from dispersoids on their boundaries compared to CubeND{001}\left and P{011}\left subgrains, with the latter experiencing the lowest drag. Subgrains at constituent particles are observed to have a growth advantage due to a lower dislocation density and higher boundary misorientation angle. The dispersoid size per subgrain boundary length increases as a function of misorientation angle. The workflow should be applicable to other alloy systems where there is a need for analysis correlating grains and grain boundaries with secondary phases smaller than the applied EBSD step size but resolvable by BSE imaging

Aggregation model for curtailable generation and sheddable loads

This paper shows modelling developed during the first year of the SmartNet project. In particular, it presents a mathematical model for aggregation of curtailable generation and sheddable loads. The model determines the quantity and the cost of the flexibility provided by the flexible resources based on their physical and dynamic behaviours. The model also proposes a bidding strategy in order to translate the aggregated behaviour into market bids

An introduction to Lie group integrators -- basics, new developments and applications

We give a short and elementary introduction to Lie group methods. A selection of applications of Lie group integrators are discussed. Finally, a family of symplectic integrators on cotangent bundles of Lie groups is presented and the notion of discrete gradient methods is generalised to Lie groups

Numerical Methods for Optical Interference Filters

We present the physics behind general optical interference filters and the design of dielectric anti-reflective filters. These can be anti-reflective at a single wavelength or in an interval. We solve the first case exactly for single and multiple layers and then present how the second case can be solved through the minimisation of an objective function. Next, we present several optimisation methods that are later used to solve the design problem. Finally, we test the different optimisation methods on a test problem and then compare the results with those obtained by the OpenFilters computer programme

In-situ observations of dislocation recovery and low angle boundary formation in deformed aluminium

An experimental study of the recovery of dislocations and low angle boundary formation in aluminium is presented. By combining in-situ annealing with orientation mapping in the transmission electron microscope, maps of geometrically necessary dislocation estimates derived from orientation measurements and subgrain formation can be quantitatively analysed. A thin foil of a commercially pure aluminium alloy cold-rolled to a true strain of ε = 2.3 and annealed in-situ in four steps of increasing temperatures from 170 °C to 560 °C was studied. An increase in the subgrain size and low angle boundary misorientation was accompanied by a halving of the dislocation density from 1.2 × 1016 m−2 to 0.6 × 1016 m−2. Limited boundary migration was observed and the increased subgrain size was attributed to the dissolution of dislocations within the low angle boundaries upon annealing

Precipitation and recrystallisation in Al-Mn-Zr with and without Sc”, To be published (Conference proceedings ICSMA13

Abstract The transition elements Zr and Mn form dispersoids in aluminium alloys and are extensively used in order to control the microstructure during heat exposure, i.e. to prevent recrystallisation and strength loss after deformation hardening. Additions of Zr have, through the formation of a dense dispersion of coherent and metastable Al 3 Zr-dispersoids, proved to be beneficial in many alloys. In the Al-Mn-Zr-alloy studied here, TEM-investigations of homogenised material have shown that these phases are heterogeneously distributed. However, by adding Sc a high density of small and well-dispersed Al 3 (Sc, Zr)-dispersoids is obtained during homogenisation, and the Sc-containing variant consequently displays a higher recrystallisation resistance after cold rolling

Precipitation and recrystallisation in Al-Mn-Zr with and without Sc”, To be published (Conference proceedings ICSMA13

Abstract The transition elements Zr and Mn form dispersoids in aluminium alloys and are extensively used in order to control the microstructure during heat exposure, i.e. to prevent recrystallisation and strength loss after deformation hardening. Additions of Zr have, through the formation of a dense dispersion of coherent and metastable Al 3 Zr-dispersoids, proved to be beneficial in many alloys. In the Al-Mn-Zr-alloy studied here, TEM-investigations of homogenised material have shown that these phases are heterogeneously distributed. However, by adding Sc a high density of small and well-dispersed Al 3 (Sc, Zr)-dispersoids is obtained during homogenisation, and the Sc-containing variant consequently displays a higher recrystallisation resistance after cold rolling